Sequential pre-spectra

A simple first definition is to define a spectrum E\mathbf{E} to be a sequence of pointed spaces (En)n∈ℕ(E_n)_{n\in\mathbb{N}} together with structure maps ΣEn→En+1\Sigma{}E_n\to{}E_{n+1} (where Σ\Sigma denotes the reduced suspension). See at model structure on sequential spectra.

There are various conditions that can be put on the spaces EnE_n and the structure maps, for example if the spaces are CW-complexes and the structure maps are inclusions of subcomplexes, the spectrum is called a CW-spectrum.

Without any condition, this is just called a spectrum, or sometimes a pre-spectrum. In order to distinguish from various other richer definitions (such as coordinate-free spectra, one also speaks of sequential spectra).

Ω\Omega-spectra

If Ω\Omega denotes the loop space functor on the category of pointed spaces, we know that Σ\Sigma is left adjoint to Ω\Omega. In particular, given a spectrum E\mathbf{E}, the structure maps can be transformed into maps En→ΩEn+1E_n\to\Omega{}E_{n+1}. If these maps are isomorphisms (depending on the situation it can be weak equivalences or homeomorphisms), then E\mathbf{E} is called an Ω\Omega-spectrum.

The idea is that E0E_0 contains the information of E\mathbf{E} in dimensions k≥0k\ge 0, E1E_1 contains the information of E\mathbf{E} in k≥−1k\ge -1 (but shifted up by one, so that it is modeled by the ≥0\ge 0 information in the space E1E_1), and so on.

Coordinate-free spectra

Combinatorial spectra

There might be a type of categorical structure related to a spectrum in the same way that ∞\infty-categories are related to ∞\infty-groupoids. In other words, it would contain kk-cells for all integers kk, not necessarily invertible. Some people have called this conjectural object a ZZ-category. “Connective” ZZ-categories could perhaps then be identified with stably monoidal ∞\infty-categories.

General context

Examples

Properties

Stabilization

In direct analogy to how topological spaces form the archetypical example, Top, of an (∞,1)-category, spectra form the archetypical example Sp(Top)Sp(Top) of a stable (∞,1)-category. In fact, there is a general procedure for turning any pointed(∞,1)-categoryCC into a stable (∞,1)(\infty,1)-category Sp(C)Sp(C), and doing this to the category Top*Top_* of pointed spaces yields Sp(Top)Sp(Top).